Gaussian fluctuations of spatially inhomogeneous polymers

Inhomogeneous polymers, such as partially cofilin-bound actin filaments, play an important role in various natural and biotechnological systems. At finite temperatures, inhomogeneous polymers exhibit non-trivial thermal fluctuations. More broadly, these are relatively simple examples of fluctuations in spatially inhomogeneous systems, which are less understood compared to their homogeneous counterparts. Here we develop a statistical theory of torsional, extensional and bending Gaussian fluctuations of inhomogeneous polymers (chains), where the inhomogeneity is an inclusion of variable size and stiffness, using both continuum and discrete approaches. First, we analytically calculate the complete eigenvalue and eigenmode spectra within a continuum field theory. In particular, we show that the wavenumber inside and outside of the inclusion is nearly linear in the eigenvalue index, with a nontrivial coefficient. Second, we solve the corresponding discrete problem and highlight fundamental differences between the continuum and discrete spectra. In particular, we demonstrate that above a certain wavenumber the discrete spectrum changes qualitatively and discrete evanescent eigenmodes, which do not have continuum counterparts, emerge. The implications of these differences are explored by calculating fluctuation-induced forces associated with free-energy variations with either the inclusion properties (e.g. inhomogeneity formed by adsorbing molecules) or with an external geometric constraint. The former, which is the fluctuation-induced contribution to the adsorbing molecule binding force, is shown to be affected by short wavelengths and thus cannot be calculated using the continuum approach. The latter, on the other hand, is shown to be dominated by long wavelength shape fluctuations and hence is properly described by the continuum theory.